Finite Group Databasebeta

← #26220520

Subgroups — #26220520

Order 12032.

798 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#10625337C211yesmaximal, characteristic
2#20245866C211yesmaximal, characteristic
3#20245862C211yesmaximal, characteristic
4#20244256C212yesmaximal
5#20244125C212yesmaximal
6#20244079C212yesmaximal
7#20244075C212yesmaximal
8C4×#13093955C211yesmaximal, characteristic
9C4×#13093949C211yesmaximal, characteristic
10C4×#13093934C211yesmaximal, characteristic
11C4×#103×C47C211yesmaximal, characteristic
12C22×#10625337C2211yescharacteristic
13C2×C4×Dic94C2211yescharacteristic
14C4×#10625337C418yescomplemented
15C2×#13093957C2211yescharacteristic
16C2×#13093955C2212yes—
17C2×#13093949C2212yes—
18C2×#13093948C2211yescharacteristic
19C2×#13093947C2211yescharacteristic
20C2×#13093934C2212yes—
21C2×#13093934C2212yes—
22C2×C4×C47⋊C8C2211yescharacteristic
23#15689030C2212yes—
24#15689028C2212yes—
25#15689022C2212yes—
26#15689019C2212yes—
27#15689009—22no—
28#15689005—22no—
29C4×Dic47⋊C4—21no—
30C4×C4⋊Dic47—21no—
31C4×C4⋊Dic47C2211yescharacteristic
32#15688969—21no—
33#15688969—21no—
34C42×Dic47C2211yescharacteristic
35#15688886C2212yes—
36#15688885C2212yes—
37C2×C4×Q8×C47C2211yescharacteristic
38C2×#103×C47C2212yes—
39C2×C4×C8×C47C2211yescharacteristic
40C4×C4⋊C4×C47C2211yescharacteristic
41C47×#340C2212yes—
42C22×Dic94C2311yescharacteristic
43C2×#10625337—24no—
44C2×#10625337C2×C414yes—
45C22×C47⋊C8C2311yescharacteristic
46C22×C47⋊C8C2311yescharacteristic
47C4×Dic94—22no—
48C4×Dic94C2×C414yes—
49C2×C4⋊Dic47—22no—
50C2×C4⋊Dic47C2312yes—
51C2×C4⋊Dic47C2311yescharacteristic
52C2×Dic47⋊C4—21no—
53C2×Dic47⋊C4—22no—
54C2×C4×Dic47—21no—
55C2×C4×Dic47—21no—
56C2×C4×Dic47D411yescharacteristic
57C2×C4×Dic47D411yescharacteristic
58C2×C4×Dic47D412yes—
59C2×C4×Dic47C2312yes—
60#13093957C2×C414yes—
61#13093955—24no—
62#13093949—24no—
63#13093948C2×C414yes—
64#13093947C2×C414yes—
65#13093934—24no—
66#13093934—24no—
67C4×C47⋊C8C2×C414yes—
68C47⋊#95—21no—
69C47⋊#95—22no—
70C47⋊#95—22no—
71C47⋊#95—22no—
72C47⋊#95—22no—
73C47⋊#95—22no—
74C47⋊#95—22no—
75C22×Q8×C47C2311yescharacteristic
76C22×C8×C47C2312yes—
77C4×Q8×C47—22no—
78C4×Q8×C47C2×C414yes—
79C2×C4⋊C4×C47—21no—
80C2×C4⋊C4×C47C2312yes—
81C2×C4⋊C4×C47C2311yescharacteristic
82C2×C42×C47—21no—
83C2×C42×C47—21no—
84C2×C42×C47C2311yescharacteristic
85#103×C47—24no—
86C4×C8×C47—22no—
87#95×C47—21no—
88C2×Q8×C47—21no—
89C2×Q8×C47—22no—
90C2×Q8×C47—22no—
91C2×Q8×C47C22×C412yes—
92C22×C4×C47—21no—
93C22×C4×C47—21no—
94C22×C4×C47—22no—
95C22×C4×C47C2×D411yescharacteristic
96C22×C4×C47C2×D411yescharacteristic
97C22×C4×C47C2411yescharacteristic
98C22×Dic47—21no—
99C22×Dic47—21no—
100C22×Dic47—22no—
101C22×Dic47C2×D412yes—
102C22×Dic47C2×D412yes—
103C2×Dic94—22no—
104C2×Dic94—22no—
105C2×Dic94—21no—
106C2×Dic94C22×C412yes—
107C2×C8×C47—22no—
108C2×C8×C47—22no—
109C2×C8×C47#4114yes—
110C4⋊C4×C47—24no—
111C4⋊C4×C47—21no—
112C4⋊C4×C47—22no—
113C4⋊C4×C47—22no—
114C4⋊C4×C47C22×C412yes—
115C42×C47—24no—
116C42×C47—42no—
117C42×C47C22×C412yes—
118C42×C47#4112yes—
119#10625337—28no—
120C4⋊Dic47—21no—
121C4⋊Dic47—22no—
122C4⋊Dic47—22no—
123C4⋊Dic47—44no—
124C4⋊Dic47C22×C412yes—
125Dic47⋊C4—24no—
126Dic47⋊C4—44no—
127C4×Dic47—24no—
128C4×Dic47—22no—
129C4×Dic47—22no—
130C4×Dic47—22no—
131C4×Dic47—22no—
132C4×Dic47—42no—
133C4×Dic47—42no—
134C2×C47⋊C8—22no—
135C2×C47⋊C8C22×C412yes—
136C2×C47⋊C8C22×C412yes—
137C2×C47⋊C8#4114yes—
138C23×C47C22×D411yescharacteristic
139Q8×C47—22no—
140Q8×C47—22no—
141Q8×C47—42no—
142C2×C4×C47—22no—
143C2×C4×C47—22no—
144C2×C4×C47—22no—
145C2×C4×C47—42no—
146C2×C4×C47—42no—
147C2×C4×C47—42no—
148C2×C4×C47—42no—
149C2×C4×C47—42no—
150C2×C4×C47C4×D412yes—
151C2×C4×C47C4×D412yes—
152C2×C4×C47#13712yes—
153C2×C4×C47#13712yes—
154C2×C4×C47C23×C411yescharacteristic
155C2×C4×C47C2×#4112yes—
156C2×C4×C47C2×#4111yescharacteristic
157C2×C4×C47C2×#4112yes—
158C2×Dic47—22no—
159C2×Dic47—22no—
160C2×Dic47—22no—
161C2×Dic47—22no—
162C2×Dic47—22no—
163C2×Dic47—22no—
164C2×Dic47—22no—
165C2×Dic47—22no—
166C2×Dic47—42no—
167C2×Dic47—42no—
168C2×Dic47—42no—
169C2×Dic47—42no—
170C2×Dic47—42no—
171Dic94—22no—
172Dic94—22no—
173Dic94—42no—
174C8×C47—24no—
175C47⋊C8—22no—
176C47⋊C8#11714yes—
177#21278—471nomaximal, Sylow
178C22×C47C2×C4×D411yescharacteristic
179C22×C47#53212yes—
180C22×C47C2×#13711yescharacteristic
181C22×C47C2×#13711yescharacteristic
182C22×C47#57512yes—
183C4×C47—22no—
184C4×C47—42no—
185C4×C47—42no—
186C4×C47—82no—
187C4×C47#44114yes—
188C4×C47C2×#11711yescharacteristic
189C4×C47C4×#4111yescharacteristic
190C4×C47#52912yes—
191Dic47—42no—
192Dic47—42no—
193Dic47—42no—
194Dic47—42no—
195Dic47—82no—
196C2×C4×Q16—471no—
197#2295—471no—
198#2295—471no—
199#1971—472no—
200#1943—472no—
201#1935—472no—
202#1935—472no—
203C4×#106—471no—
204C4×#103—471no—
205C4×#103D4711yescharacteristic
206C4×#97—471no—
207C2×C47#229311yescharacteristic
208C2×C47C2×#44111yescharacteristic
209C2×C47#294311yescharacteristic
210C2×C47#317212yes—
211C2×C47#317212yes—
212C22×Q16—471no—
213C2×C4×Q8—471no—
214C2×C4×Q8C2×D4711yescharacteristic
215C4×Q16—478no—
216C2×#107—471no—
217C2×#106—472no—
218C2×#103—471no—
219C2×#103—471no—
220C2×#103—472no—
221C2×#103C2×D4712yes—
222C2×#97—472no—
223C2×#97—472no—
224C2×C4×C8—471no—
225C2×C4×C8C2×D4711yescharacteristic
226#391—472no—
227#391—472no—
228#389—472no—
229#389—472no—
230#387—942no—
231#387—942no—
232C4×C4⋊C4—471no—
233C4×C4⋊C4—941no—
234C4×C4⋊C4—941no—
235C4×C4⋊C4C2×D4711yescharacteristic
236#376—941no—
237#376—941no—
238C43—471no—
239#340—472no—
240#340—472no—
241#340C2×D4712yes—
242C47#2127811yescharacteristic, complemented, Sylow
243C22×Q8—471no—
244C22×Q8C22×D4711yescharacteristic
245C2×Q16—474no—
246C2×Q16—944no—
247C22×C8—471no—
248C22×C8—471no—
249C22×C8C22×D4712yes—
250C4×Q8—22no—
251C4×Q8—474no—
252C4×Q8—942no—
253C4×Q8C4×D4714yes—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—471no—
256C2×C4⋊C4—472no—
257C2×C4⋊C4—941no—
258C2×C4⋊C4—942no—
259C2×C4⋊C4—942no—
260C2×C4⋊C4C22×D4712yes—
261C2×C4⋊C4C22×D4711yescharacteristic
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—471no—
265C2×C42—471no—
266C2×C42—472no—
267C2×C42—472no—
268C2×C42—941no—
269C2×C42—941no—
270C2×C42C22×D4711yescharacteristic
271#107—474no—
272#106—944no—
273#103—24no—
274#103—474no—
275#103—474no—
276#103—944no—
277#97—944no—
278#97—944no—
279C4×C8—22no—
280C4×C8—474no—
281#95—21no—
282#95—942no—
283#95—942no—
284#95—942no—
285#95—942no—
286#95—942no—
287#95—942no—
288#95—941no—
289C2×Q8—21no—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—472no—
293C2×Q8—941no—
294C2×Q8—942no—
295C2×Q8—942no—
296C2×Q8C2×C4×D4712yes—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—472no—
301C22×C4—472no—
302C22×C4—941no—
303C22×C4—941no—
304C22×C4—942no—
305C22×C4D4×D4711yescharacteristic
306C22×C4D4×D4711yescharacteristic
307C22×C4C23×D4711yescharacteristic
308Q16—948no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—472no—
312C2×C8—474no—
313C2×C8—472no—
314C2×C8—942no—
315C2×C8#1062535014yes—
316C4⋊C4—24no—
317C4⋊C4—21no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—472no—
321C4⋊C4—944no—
322C4⋊C4—941no—
323C4⋊C4—942no—
324C4⋊C4—942no—
325C4⋊C4—1884no—
326C4⋊C4—1884no—
327C4⋊C4C2×C4×D4712yes—
328C42—24no—
329C42—42no—
330C42—944no—
331C42—942no—
332C42—942no—
333C42—942no—
334C42—942no—
335C42—1882no—
336C42—1882no—
337C42C2×C4×D4712yes—
338C42#1062535212yes—
339C23C2×D4×D4711yescharacteristic
340Q8—22no—
341Q8—22no—
342Q8—42no—
343Q8—942no—
344Q8—942no—
345Q8—1882no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—942no—
355C2×C4—942no—
356C2×C4—942no—
357C2×C4—942no—
358C2×C4—942no—
359C2×C4—942no—
360C2×C4—942no—
361C2×C4—942no—
362C2×C4—1882no—
363C2×C4—1882no—
364C2×C4—1882no—
365C2×C4—1882no—
366C2×C4—1882no—
367C2×C4(C22⋊D47)⋊C412yes—
368C2×C4(C22⋊D47)⋊C412yes—
369C2×C4#1309405312yes—
370C2×C4#1309405612yes—
371C2×C4C22×C4×D4711yescharacteristic
372C2×C4#41×D4712yes—
373C2×C4C2×#1062535211yescharacteristic
374C2×C4C2×#1062535012yes—
375C8—24no—
376C8—474no—
377C8—942no—
378C22C2×(C22⋊D47)⋊C411yescharacteristic
379C22#1568967012yes—
380C22C2×#1309405311yescharacteristic
381C22C2×#1309405611yescharacteristic
382C22#1568992012yes—
383C4—22no—
384C4—42no—
385C4—42no—
386C4—82no—
387C4—3762no—
388C4—1882no—
389C4—1882no—
390C4—1882no—
391C4—1882no—
392C4#1568922714yes—
393C4#117×D4711yescharacteristic
394C4C4×#1062535011yescharacteristic
395C4#1568965312yes—
396C2#2024585111yescharacteristic
397C2C2×#1568922711yescharacteristic
398C2#2025218411yescharacteristic
399C2#2025399012yes—
400C2#2025399212yes—
subgroupG/H
centerC22×C4D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC4×#103×C47C2
socle, prime radicalC23×C47C22×D4
cosocleC22×C4×C47C24
hypercenterC4×#103D47
hypocenterC47#21278
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C4×#103

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
14. C4×#10625337C4semidirect
242. C47#21278semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C4×#106253372
#202458662
#202458622
#202442562
#202441252
#202440792
#202440752
C4×#130939552
C4×#130939492
C4×#130939342
C4×#103×C472
#2127847